Physical quantity,product of vector and lami's theorem

VECTOR AND SCALARS

Physical Quantity

May have numerical value, units and any specified direction.

Note:-  Some physical quantities only having numerical value not specified direction.

Ex:-  Refractive index, strain etc.

We can say any physical quantity Must/may have numerical value(n), unit (u) and specified direction.

PHYSICAL QUANTITY (N + U + DIRECTION) MAY/MUST HAVE

Note:

Scalars or zero order tensors: Any physical quantity have only one component.

Vectors or first order tensors: Any physical Quantity have component greater then one but less than or equal to four.

REPRESENTATION OF VECTOR:

Types of vectors

(1) Polar Vectors:

Vectors related to linear motion of any object Ex. Displacement, velocity, Force etc.

(2) Axial vectors:

Vector represent rotational effect and are always along the axis of rotation Ex: Angular velocity, torque, angular momentum etc.

(3)  Null vector or zero vectors:

Vector whose magnitude is zero and direction in determinant.

(4) Unit vectors:

Vector having magnitude equals to one (unity) but must be some specified direction representation of unit vector – , A  

 

STANDARD UNIT VECTOR

These are

Direction should be on your copy

UNDERSTANDING OF UNIT VECTORS

1 –D Motion: Motion along x or y or z

A car is moving with 60 k/m toward or due east then speed (scalar) = 60 km/h

Velocity (Vector) = 60 km/h

2 –D Motion:

A car is moving with 60 k/m due north – east then

 

 

2 –D vector

 

With x axis or with horizontal

 

With Y-axis or with vertical

3 –D Vector

 DIRECTION

 

With x-axis

 

With y-axis

 

With z-axis

ADDITION AND SUBSTRACTION OF VECTORS

Law of parallelogram

 

 

 

 

 

 

 

 

 

 

 

 

            RESOLVING OF ANY VECTOR

Example. Two equal vector have a resultant equal to either of the two. The angle between them is                                                                                                                    

(a) 90^o                                                      (b) 60^o

(c) 120^o                                                    (d) 0^o

Solution: By using expression R² = P² + Q² + 2PQ cosa

 

x² = 2x² (1 + cosa)

cos a = -1/2 = a = 120^o                                                                             Answer is (c)

Example: Two vector having equal magnitude of x units acting at an angle of 45⁰ have resultant   units the value of x is                                                             

(a) 0                                        (b) 1

(c)                                       (d)

Solution: Using the expression R² = P² + Q² + 2PQ cosµ

 

 

 

 

 

Þ x² =1                                                                    Ans (b) Þ x = 1 Ans (b)

 

 

(a) 0°                                                   (b) 180°

(c) 90°                                                 (d) 120°

 

 

= 4PQ cos q = 0

   Cos q = 0

   q = 90^o                                                                                   Answer is (c)

          LAMI’S THEOREM